Elements of Formal Semantics - Universiteit Utrechtyoad/efs/EFS-ch2-online.pdf · Elements of Formal Semantics introduces some of the foundational concepts, principles and techniques

Elements of Formal Semantics An Introduction to the Mathematical Theory of Meaning in Natural Language

Yoad Winter

Open Access Materials: Chapter 2

� Edinburgh University Press, 2016

See webpage below for further materials and information:

http://www.phil.uu.nl/~yoad/efs/main.html

Elements of Formal Semantics introduces some of the foundational

concepts, principles and techniques in formal semantics of natural

language. It is intended for mathematically-inclined readers who have

some elementary background in set theory and linguistics. However, no

expertise in logic, math, or theoretical linguistics is presupposed. By way

of analyzing concrete English examples, the book brings central concepts

and tools to the forefront, drawing attention to the beauty and value of the

mathematical principles underlying linguistic meaning.

March 8, 2016 Time: 03:51pm chapter2.tex

CHAPTER 2

MEANING AND FORM

This chapter introduces some of the key notions about the analysis ofmeaning in formal semantics. We focus on entailments: relationsbetween premises and valid conclusions expressed as naturallanguage sentences. Robust intuitions about entailment aredistinguished from weaker types of reasoning with language. Speakerjudgments on entailments are described usingmodels: abstractmathematical structures, which emanate from semantic analyses ofartificial logical languages. Model-theoretical objects are directlyconnected to syntactic structures by applying a general principle ofcompositionality. We see how this principle helps to analyze cases ofstructural ambiguity and to distinguish them from other cases ofunder-specification.

What do dictionaries mean when they tell us that semantics is “thestudy of meaning”? The concept that people intuitively refer to as“meaning” is an abstraction inspired by observing how we use lan-guage in everyday situations. However, we use language for manydifferent purposes, and those various usages may inspire conceptionsof meaning that are radically different from one another. We cannotreasonably expect a theory of meaning to cover everything that peopledo with their languages. A more tractable way of studying meaning isby discerning specific properties of language use that are amenable toscientific investigation. These aspects of language use, if stable acrossspeakers and situations, will ultimately guide us toward a theory oflanguage “meaning”.

ENTAILMENTOne of the most important usages of natural language is for everydayreasoning. For example, let us consider sentence (2.1):

(2.1) Tina is tall and thin.

12


MEANING AND FORM 13

From this sentence, any English speaker is able to draw the conclusionin (2.2) below:

(2.2) Tina is thin.

Thus, any speaker who considers sentence (2.1) to be true, will con-sider sentence (2.2) to be true as well. We say that sentence (2.1) entails(2.2), and denote it (2.1)⇒(2.2). Sentence (2.1) is called the premise , orantecedent, of the entailment. Sentence (2.2) is called the conclusion, orconsequent.The entailment from (2.1) to (2.2) exemplifies a relation that all

English speakers will agree on. This consistency is remarkable, andall the more so since words like Tina, tall and thin are notoriouslyflexible in the way that they are used. For instance, you and I may havedifferent criteria for characterizing people as being thin, and thereforedisagree on whether Tina is thin or not. We may also disagree on theidentity of Tina. You may think that Tina in sentences (2.1) and (2.2)is Tina Turner, while I may think that these sentences describe TinaCharles. However, we are unlikely to disagree on whether sentence(2.2) is a sound conclusion from (2.1).We noted that when sentence (2.1) is judged to be true, so is

sentence (2.2). However, the converse does not hold: (2.2) may betrue while (2.1) is not – this is the case if Tina happens to be thinbut not tall. Because of such situations, we conclude that sentence(2.2) does not entail (2.1). This is denoted (2.2) �⇒(2.1). Just as withpositive judgments on entailment, rejections of entailment are alsooften uniform across speakers and circumstances of use. Therefore,we consider both positive and negative judgments on entailment asimportant empirical evidence for semantic theory.When studying simple entailments, we often pretend that our lan-

guage vocabulary is very small. Still, as soon as our vocabulary hassome simple adjectives and proper names, we can easily find entail-ments and non-entailments by looking at their different combinationswith words like and, or, is and not. For instance:

(2.3) a. Tina is tall, and Ms. Turner is not tall⇒ Tina is not Ms.Turner.

b. Tina is tall, and Tina is not Ms. Turner �⇒ Ms. Turner isnot tall.


14 ELEMENTS OF FORMAL SEMANTICS

(2.4) a. Ms. Turner is tall, and Tina is Ms. Turner or Ms. Charles⇒ Tina is tall or Tina is Ms. Charles.

b. Ms. Turner is tall, and Tina is Ms. Turner or Ms. Charles�⇒ Tina is tall.

The examples above may look unsurprising for anyone who is familiarwith philosophical or mathematical logic. Indeed, similar entailmentsin natural language have inspired well-known logical formalisms likePropositional Logic and Predicate Logic. Readers may therefore won-der: don’t the entailments above demonstrate puzzles that were solvedlong ago by logicians? The answer is “yes and no”. Sure enough, theseentailments can be translated to well-understood logical questions.However, logic does not traditionally focus on the details of the trans-lation procedure from ordinary language to logical languages. Thistranslation step is not “pure logic”: it also involves intricate questionsabout the sounds and the forms of human languages, and about thenature of semantic concepts in the human mind. Consequently, inmodern cognitive science, the study of entailment in natural languageis not the sanctuary of professional logicians. Entailment judgmentsbring to the forefront a variety of questions about language that arealso of primary concern for linguists, computer scientists, psychol-ogists and philosophers. For instance, let us consider the followingentailments:

(2.5) a. Sue only drank half a glass of wine⇒ Sue drank less thanone glass of wine.

b. A dog entered the room⇒ An animal entered the room.c. John picked a blue card from the pack ⇒ John picked a

card from the pack.

The entailments in (2.5) illustrate different aspects of language: mea-sures and quantity in (2.5a); word meaning relations in (2.5b); adjec-tivemodification in (2.5c). These kinds of entailment are very commonin natural language, but they were not systematically treated in classicallogic. By studying the whole spectrum of entailments in ordinarylanguage, formal semantics addresses various aspects of linguisticphenomena and their connections with human reasoning. Ideas from


MEANING AND FORM 15

logic are borrowed insofar as they are useful for analyzing naturallanguage semantics. More specifically, later in this book we adoptconcepts from type theory and higher-order logics that have provedespecially well suited for studying entailment in natural language. Aswe shall see, incorporating these concepts allows formal semantics todevelop important connections with theories about sentence structureand word meaning.Among the phenomena of reasoning in language, entailment is

especially central because of its remarkable stability. In other instancesof reasoning with natural language, conclusions are not fully sta-ble, since they may rely on implicit assumptions that emanate fromcontext, world knowledge or probabilistic principles. These lead tomeaning relations between sentences that are often fuzzier and lessregular than entailments. Consider for instance the following twosentences:

(2.6) Tina is a bird.

(2.7) Tina can fly.

Sentence (2.7) is a likely conclusion from (2.6), and most speakerswill not hesitate too much before drawing it. However, upon somereflection we can come up with many situations in which sentence(2.6) truthfully holds without supporting the conclusion in (2.7).Think of young birds, penguins, ostriches, or birds whose wings arebroken. Thus, in many natural discourses speakers may accept (2.6)while explicitly denying (2.7):

(2.8) Tina is a bird, but she cannot fly, because ... (she is too young tofly, a penguin, an ostrich, etc.).

We classify the inferential relation between sentences like (2.6) and(2.7) as defeasible, or cancelable, reasoning. By contrast, entailmentsare classified as indefeasible reasoning: all of the assumptions that areneeded in order to reach the conclusion of an entailment are explicitlystated in the premise. For instance, the entailment (2.1)⇒(2.2) cannotbe easily canceled by adding further information to the premise.



A discourse like (2.9) below that tries to contradict sentence (2.2) afterasserting (2.1) will normally be rejected as incoherent.

(2.9) #Tina is tall and thin, but she is not thin.

A speaker who wishes to support this incoherent line of reasoningwould need to resort to self-contradictory or otherwise counter-communicative arguments like “because I am lying when saying thatTina is tall and thin”, or “because I am not using English in the ordi-nary way”. The incoherence of (2.9) is marked by ‘#’. Sentence (2.9) isintelligible but nonsensical: its communicative value is dubious. Thissort of incoherent, contradictory sentence should be distinguishedfrom ungrammaticality. The latter notion is reserved for strings ofwords that clearly do not belong to natural language, e.g. is and tallTina thin. Such ungrammatical strings are standardly marked by ‘*’.Taking stock, we adopt the following notion of entailment:

Given an indefeasible relation between two natural language sen-tences S1 and S2, where speakers intuitively judge S2 to be truewhenever S1 is true, we say that S1 entails S2, and denote it S1⇒S2.

Just as intuitive judgments about sentence grammaticality have be-come a cornerstone in syntactic theory, intuitions about entailmentsbetween sentences are central for natural language semantics. As inother linguistic domains, we aim to build our semantic theory onjudgments that do not rely on training in linguistics, logic or otherscholarly disciplines. Entailments that robustly appear in ordinaryreasoning give us a handle on common semantic judgments aboutlanguage.Entailments between sentences allow us to define the related notion

of equivalence. For instance, the sentence (2.1)=Tina is tall andthin and the sentence S=Tina is tall and Tina is thin are classi-fied as equivalent, because they entail each other. We denote thisequivalence (2.1)⇔S . For more examples of equivalent sentences seeExercise 4. Another classical semantic notion is contradiction, whichwas lightly touched upon in our discussion of sentence (2.9) above. SeeExercise 7 for some elaboration on contradictions and their relationsto entailment.


MEANING AND FORM 17

MODELS AND THE TRUTH-CONDITIONALITYCRITERION

With this background on entailments in natural language, let us nowsee how formal semantics accounts for them. As mentioned above,formal semantics relies on some central principles from traditionalphilosophical and mathematical logic. Most versions of formal seman-tics account for entailments using theoretical structures that are calledmodels. Models are mathematical abstractions that we construct anduse as descriptions of hypothetical situations. We call these situations“hypothetical” because they do not necessarily correspond to actualsituations in the world. Some of our models may agree with howwe look at the world, but some of them will also describe situationsthat are purely imaginary. For instance, the models that we use foranalyzing the sentence Tina is thin will describe situations in whichTina is thin, as well as situations where she is not thin. If you knowa woman called Tina and you think that she is thin, you will considerthe first models as closer to reality than the others. However, this isirrelevant for our purposes. For the sake of our semantic analysis weconsider all the models that we construct as hypothetical. As such, theyare all equal.In order to encode hypothetical situations in models, we let models

link words to abstract mathematical objects. For instance, since wewant our models to describe situations in relation to the word Tina, welet each model contain some or other abstract entity that is associatedwith this word. Similarly, when we analyze the words tall and thin, wealso let our models associate these adjectives with abstract objects. Inthis chapter we let models link adjectives to sets of entities. Thus, ineach model we include a set of entities that is associated with the wordtall. These are the abstract entities in that model that are consideredtall in the hypothetical situation that the model describes. Similarly,each model associates the adjective thin with the set of entities that areconsidered thin in the situation.In addition to dealing with words, models also treat complex ex-

pressions: phrases and sentences that are made up of multiple words.For example, let us consider the complex phrase tall and thin. Justlike we did in the case of simple adjective words, we let each modelassociate this phrase with a set of entities. These are the entities thatare considered to be tall and thin in the hypothetical situation that the



model describes. Other words, phrases and sentences are associatedwith all sorts of abstract mathematical objects. The words, phrasesand sentences that we treat are collectively referred to as expressions.In each of our models, we associate abstract objects with all theexpressions that we treat.Summarizing, we state our general conception of models as follows:

A model is an abstract mathematical structure that we constructfor describing hypothetical situations. Models are used for analyzingnatural language expressions (words, phrases and sentences) by asso-ciating them with abstract objects.

Associating language expressions with abstract objects is part andparcel of a model definition. For instance, one of the models that weuse, call it M, may associate the word Tina with some abstract entitya. In technical terms, we say that in the model M, the word Tinadenotes the entity a. In all the models that we study in this chapter,the name Tina denotes some or other entity, and the adjective talldenotes some or other set of entities. Given a particular model M,we refer to those denotations as [[Tina]]M and [[tall]]M , respectively.Similarly, [[tall and thin]]M is the denotation of the phrase tall andthin in the model M. In general, we adopt the following notationalconvention:

Let exp be a language expression, and let M be a model. We write[[exp]]M when referring to the denotation of exp in the model M.

To have a more concrete view on models and denotations, let usconsider Figure 2.1. This figure describes two models, each of themcontaining three entities: a, b and c. In model M1, the name Tinadenotes the entity a, and the adjective thin denotes the set {a, b}. Inmodel M2, the denotation of Tina is again the entity a, but this time,the set denotation of thin is the set {b, c}. We formally write it asfollows:

[[Tina]]M1 = a [[thin]]M1 = {a, b}[[Tina]]M2 = a [[thin]]M2 = {b, c}


MEANING AND FORM 19

M1 M2words

Tina

thin

a

b

c

words

Tina

thin

a

b

c

Figure 2.1 Models map words and other expressions to abstractmathematical objects. M1 and M2 are models with an entity denotationof Tina and a set denotation of thin. The arrows designate the mappingsfrom the words to their denotations, which are part of the modeldefinition.

Figure 2.1 only illustrates the assignment of denotations to simplewords. However, as mentioned above, models are used in orderto assign denotations to all expressions that we analyze, includingcomplex expressions that are made of multiple words. In particular,models specify denotations for sentences. There are various ideas aboutwhat kinds of abstract objects sentences should denote. In most of thisbook, we follow the traditional assumption that sentences denote thetwo abstract objects known as truth-values, which are referred to as‘true’ and ‘false’. In more technical notation, we sometimes write ‘�’for ‘true’ and ‘⊥’ for ‘false’. Yet another convention, which is mostconvenient for our purposes, is to use the number 1 for ‘true’ and thenumber 0 for ‘false’.Models assign truth-value denotations to sentences on the basis of

the denotations they assign to words. For instance, the way we usemodels such as M1 and M2 in Figure 2.1 respects the intuition thatthe sentence Tina is thin is true in M1 but false in M2. Thus, we willmake sure that M1 and M2 satisfy:

[[Tina is thin]]M1 = 1 [[Tina is thin]]M2 = 0

As we move on further in this chapter, we see how this analysis isformally obtained.The truth-value denotations that models assign to sentences are the

basis for our account of entailment relations. Let us return to theentailment between the sentence Tina is tall and thin (=(2.1)) and thesentence Tina is thin (=(2.2)). When discussing this entailment, we



informally described our semantic judgment by saying that wheneversentence (2.1) is true, sentence (2.2) must be true as well. By contrast,we observed that the intuition does not hold in the other direction:when (2.2) is true, sentence (2.1) may be false. For this reason weintuitively concluded that sentence (2.2) does not entail (2.1). Whenanalyzing (non-)entailment relations, we take into account these pre-theoretical intuitions about ‘truth’ and ‘falsity’. We analyze an entail-ment S1⇒S2 by introducing the following requirement: if a modellets S1 denote true, it also lets S2 denote true. When truth-values arerepresented numerically, this requirement means that if a model letsS1 denote the number 1, it also lets S2 denote 1.Specifically, in relation to the entailment (2.1)⇒(2.2), we require

that for every model where sentence (2.1) denotes the value 1, sentence(2.2) denotes 1 as well. Another way to state this requirement is tosay that in every model, the truth-value denotation of (2.1) is less thanor equal to the denotation of (2.2). Let us see why this is indeed anequivalent requirement. First, consider models where the denotationof (2.1) is 1. In such models, we also want the denotation of (2.2)to be 1. Indeed, requiring [[(2.1)]]≤[[(2.2)]] boils down to requiringthat [[(2.2)]] is 1: this is the only truth-value for (2.2) that satisfies1≤[[(2.2)]]. Further, when we consider models where the denotationof (2.1) is 0, we see that such models trivially satisfy the requirement0≤[[(2.2)]], independently of the denotation of (2.2).To conclude: saying that [[(2.2)]] is 1 in every model where [[(2.1)]]

is 1 amounts to saying that [[(2.1)]]≤[[(2.2)]] holds in every model.Accordingly, we translate our intuitive analysis of the entailment(2.1)⇒(2.2) to the formal requirement that [[(2.1)]]≤[[(2.2)]] holds inevery model. More generally, our aim is to account for entailmentsusing the ≤ relation between truth-values in models. This leads to acentral requirement from formal semantic theory, which we call thetruth-conditionality criterion (TCC):

A semantic theory T satisfies the truth-conditionality criterion(TCC) for sentences S1 and S2 if the following two conditions areequivalent:(I) Sentence S1 intuitively entails sentence S2.(II) For all models M in T: [[S1]]M ≤ [[S2]]M.


MEANING AND FORM 21

ENTAILMENT

S1⇒ S2TCC←→

MODELS

[[S1]]M1 ≤ [[S2]]M1

[[S1]]M2 ≤ [[S2]]M2

[[S1]]M3 ≤ [[S2]]M3

. . .

Figure 2.2 The TCC matches judgments on entailment with the ≤relation. When the entailment S1⇒S2 holds, the ≤ relation is requiredto hold between the truth-value denotations of S1 and S2 in all themodels of the theory.

Table 2.1: Does x ≤ y hold?y=0 y=1

x=0 yes yesx=1 no yes

Clause (I) of the TCC postulates an entailment between sentences S1and S2. This is an empirical statement about the semantic intuitionsof native speakers. By contrast, clause (II) is a statement about ourtheory’s treatment of sentences S1 and S2: the formal models we useindeed rely on intuitive notions of ‘truth’ and ‘falsity’, but they arepurely theoretical. By imposing a connection between the empiricalclause (I) and the theoretical clause (II), the TCC constitutes anadequacy criterion for semantic theory.By way of recapitulation, Figure 2.2 illustrates how the TCC em-

ulates the intuitive relation of entailment (‘⇒’) between sentences,by imposing the ≤ relation on sentence denotations in the modelsthat our theory postulates. Table 2.1 summarizes how the requirementx ≤ y boils down to requiring that if x is 1, then y is 1 as well. Readerswho are familiar with classical logicmay observe the similarity betweenTable 2.1 and the truth table for implication. We will get back to thispoint in Chapter 5.



ARBITRARY AND CONSTANT DENOTATIONSWe have introduced the TCC as a general criterion for the empiricaladequacy of formal semantics. Obviously, we want our theory torespect the TCC for as many intuitive (non-)entailments as possible.This will be our main goal throughout this book. Let us start ourinvestigations by using the TCC to explain our favorite simple en-tailment: (2.1)⇒(2.2). We want to make sure that the models thatwe construct respect the TCC for this entailment. Thus, we need todefine whichmodels we have in our theory, and then check what truth-values each model derives for sentences (2.1) and (2.2). The modelsthat we define fix the denotations of words like Tina, tall and thin. Intechnical jargon, words are also called lexical items. Accordingly, wewill refer to the denotations of words as lexical denotations. Based onthe lexical denotations that models assign to words, we will define thetruth-values assigned to sentences containing them. To do that, let usexplicitly state the assumptions that we have so far made about ourmodels:

1. In every model M, in addition to the two truth-values 0 and 1, wehave an arbitrary non-empty set E M of the entities in M. We referto this set as the domain of entities in M. For instance, in modelsM1 and M2 of Figure 2.1, the entity domains E M1 and E M2 are thesame: in both cases they are the set {a, b, c}.

2. In any model M, the proper name Tina denotes an arbitrary entityin the domain E M (cf. Figure 2.1).

3. In any model M, the adjectives tall and thin denote arbitrary sets ofentities in E M (cf. Figure 2.1).

When the model is understood from the context, we often writeE for the domain of entities, suppressing the subscript M. We saythat the domains of entities in the models we define are ‘arbitrary’because we do notmake any special assumptions about them: any non-empty set may qualify as a possible domain of entities in some model.Accordingly, we also treat the entity denotation of Tina as an arbitraryelement of E . Whether this entity corresponds to a real-life entity likeTina Turner or Tina Charles is not our business here. We are not eveninsisting that the entity for Tina has ‘feminine’ properties, as might besuitable for a feminine English name. All we require is that in everymodel, the name Tina denotes some entity. In a similar fashion, we let


MEANING AND FORM 23

the adjectives tall and thin denote arbitrary sets in our models. Thisarbitrariness is of course an over-simplification, but it will do for ourpurposes in this book. Here we study the meanings of words only tothe extent that they are relevant for the study of entailment. Of course,much more should be said on word meanings. This is the focus ofresearch in lexical semantics, which deals with many other importantaspects of wordmeaning besides their contribution to entailments. Forsome readings on this rich domain, see some recommendations at theend of this chapter.From now on we will often use words in boldface when referring to

arbitrary denotations. For instance, by ‘tina’ we refer to the element[[Tina]] of E that is denoted by the word Tina in a given model.Similarly, ‘tall’ and ‘thin’ are shorthand for [[tall]] and [[thin]]: the setsof entities in E denoted by the words tall and thin. Putting wordsin boldface in this way is a convenience that spares us the use of thedouble brackets [[]].Whenwewant to bemore specific about themodelM, we write tinaM or [[Tina]]M .In our discussion of Figure 2.1, we noted that the sentence Tina is

thin is intuitively true in M1 but false in M2. We can now see how thisintuition is respected by our precise definition of models. To achievethat, wemake sure that the sentences Tina is thin reflects amembershipassertion. We only allow the sentence to be true in models where theentity denoted by Tina is a member of the set denoted by the adjective.Therefore, we analyze the word is as denoting a membership function.This is the function sending every entity x and set of entities A to thetruth-value 1 if x is an element of A. If x is not an element of A,the membership function sends x and A to the truth-value 0. Whenreferring to the membership function that the word is denotes, we usethe notation ‘IS’. Formally, we define IS as the function that satisfies thefollowing, for every entity x in E and every subset A of E :

IS(x, A)={1 if x ∈ A0 if x �∈ A

For example, let us reconsider the models M1 and M2 that we saw inFigure 2.1. With our new assumption on the denotation of the wordis, we now get:

In M1: [[Tina is thin]] = IS(tina, thin)= IS(a, {a, b})= 1 since a ∈ {a, b}In M2: [[Tina is thin]] = IS(tina, thin)= IS(a, {b, c})= 0 since a �∈ {b, c}



Thus, in M1 the sentence Tina is thin denotes 1, and in M2 it denotes 0,as intuitively required. More generally, in (2.10) below we summarizethe denotation that the sentenceTina is thin is assigned in everymodel:

(2.10) [[Tina is thin]]M = IS(tina, thin) ={1 if tina ∈ thin0 if tina �∈ thin

When referring to denotations, we have made a difference betweenthe font for the denotations tina, tall and thin, and the font forthe denotation IS. There is a reason for this notational difference.As mentioned, the denotations of the words Tina, tall and thin arearbitrarily chosen by our models. We have presented no semantic‘definition’ for the meaning of these words. Models are free to let thename Tina denote any of their entities. Similarly, the adjectives talland thinmay denote any of set of entities. By contrast, the denotationof the word is has a constant definition across models: in all modelswe define this denotation as the membership function. We will havemore to say about this distinction between denotations in Chapter 3.In the meantime, let us summarize our notational conventions:

Let blik be a word in a language. When the denotation [[blik]]M ofblik is arbitrary, we mark it blik. When it has a constant definitionacross models we mark it BLIK.

ANALYZING AN ENTAILMENTIn order to analyze the entailment (2.1)⇒(2.2), let us now also lookat the denotation of the sentence Tina is tall and thin. Since we let thesentenceTina is thin denote the truth-value of amembership assertion,it is only natural to analyze the sentence Tina is tall and thin in asimilar way. Thus, we want this sentence to be true if the entity tinais a member of a set denoted by the conjunction tall and thin. Butwhat should this set be? The same semantic intuitions that supportedthe entailment (2.1)⇒(2.2) can guide us to the answer. Obviously, forTina to be tall and thin, she has to be tall, and she also has to be thin.And vice versa: if Tina is tall, and if in addition she is also thin, thereis no way to avoid the conclusion that she is tall and thin. Elementaryas they are, these considerations suggest that if we are going to let the


MEANING AND FORM 25

conjunction tall and thin denote a set, it had better be the intersectionof the two sets for the adjectives tall and thin. Formally, we write:

[[tall and thin]]M = [[tall]]M ∩ [[thin]]M = tall∩ thin.Thus, we define the denotation of the word and to be the intersectionfunction over E . This is the function AND that satisfies the following,for all subsets A and B of E :

AND(A, B) = A∩ B

= the set of all members of E that are both in A and in B

Now there is also no doubt about the denotation of the sentence Tinais tall and thin. Using the same kind of membership assertion that weused for the sentence Tina is thin, we reach the following denotationfor this sentence:

(2.11) [[Tina is tall and thin]]M = IS( tina, AND(tall,thin) )

={1 if tina ∈ tall∩ thin0 if tina �∈ tall∩ thin

In words: in every given model M, the sentence Tina is tall and thindenotes the truth-value 1 if the entity tina is in the intersection of thesets tall and thin; otherwise the sentence denotes 0.We have now defined the truth-value denotations that the sentences

Tina is tall and thin and Tina is thin have in every model. These arethe truth-values specified in (2.11) and (2.10), respectively. Therefore,we can use the TCC in order to verify that our theory adequatelydescribes the entailment between the two sentences. As a matter ofset theory, the truth-value (2.10) must be 1 if the truth-value in (2.11)is 1: if the entity tina is in the intersection tall∩ thin, then, bydefinition of intersection and set membership, it is also in the set thin.This set-theoretical consideration holds for all possible denotationstina, tall and thin. Thus, it holds for all models. This means thatour assignment of denotations to sentences (2.1) and (2.2) has beensuccessful in meeting the TCC when accounting for the entailmentbetween them.At this point you may feel that the games we have been playing

with entities, sets, functions and truth-values are just restating obviousintuitions. This is perfectly true. Indeed, there is reason to feel satisfied



about it. Semantic models provide us with a general and mathemati-cally rigorous way of capturing common intuitions about entailment.A model is a small but precise description of a particular situationin which different sentences may be true or false. By specifying thedenotations of the words Tina, tall and thin, a model describes, in anabstract way, who Tina is, and what the tall entities and thin entitiesare. As we have seen, and as we shall see in more detail throughoutthis book, models also take care of more “logical” denotations forwords like and and is. This assignment of denotations to lexical itemsenables us to systematically assign denotations to complex expressions,including conjunctive phrases like tall and thin and sentences likeTina is tall and thin. If we are successful in assigning denotations tosuch complex expressions, we may be reasonably hopeful that ourstrategies will also be useful for much higher levels of hierarchicalembedding (e.g. Dylan’s description of the sad-eyed lady on page1). In fact, by defining truth and falsity in models for two simplesentences, we have been forced to dive rather deep into the meaningsof conjunction, predication, adjectives and proper names, and the waysin which they combine with each other. As we shall see in the fol-lowing chapters, much of our elementary set-theoretical maneuveringso far is valuable when tackling more advanced questions in formalsemantics.When looking at a class of models that is heterogenous enough, we

can “see”, so to speak, whether one sentence must denote 1 whenanother sentence does. Let us get a feel of what is going on in thesimple example we have been treating, by playing a little with someconcrete models. Let us consider Table 2.2, which summarizes ourassumptions so far and illustrates them concretely in the three modelsdescribed in the rightmost columns. Each of these models has the setE = {a, b, c, d} as its domain of entities. In model M1, the word Tinadenotes the entity a, and the word thin denotes the set of three entities{a, b, c}. Model M2 assigns different denotations to these words: Tinadenotes the entity b, and thin denotes the set {b, c}. In model M3, thedenotation of Tina remains the entity b, as in model M2, while theadjective thin denotes a set of three entities: {a, c, d}. Accordingly, thetruth-values in the three models for the sentence Tina is thin are 1, 1and 0, respectively. Similarly, using the assumed denotations for tall,we can also verify that the truth-values in these three models for thesentence Tina is tall and thin are 0, 1 and 0, respectively. Satisfying


MEANING AND FORM 27

Table 2.2: Denotations for expressions in the entailment (2.1)⇒(2.2).

Expression Cat. Type Abstract denotationDenotations inexample models withE = {a, b, c, d}M1 M2 M3

Tina PN entity tina a b btall A set of entities tall {b, c} {b, d} {a, b, d}thin A set of entities thin {a, b, c} {b, c} {a, c, d}tall and thin AP set of entities AND(tall, thin) {b, c} {b} {a, d}Tina is thin S truth-value IS(tina, thin) 1 1 0Tina is tall and thin S truth-value IS(tina, AND(tall, thin)) 0 1 0

Categories: PN = proper name; A = adjective; AP = adjective phrase; S = sentence.

the TCC means that the latter value must be less than or equal to theformer value, which is indeed the case.Model M1 in Table 2.2 shows that the TCC is also met for the

non-entailment (2.2) �⇒(2.1). This model makes sentence (2.2) truewhile making (2.1) false. This means that our theory respects therequirement in the TCC that, if an entailment is missing, then atleast one model does not satisfy the ≤ relation between the truth-values of the two sentences in question. In formula, model M1 satis-fies [[(2.2)]]M1 �≤[[(2.1)]]M1 . Furthermore, model M1 also respects ourintuition of why an entailment is absent in this case. As pointed outabove, if somebody tried to convince you that Tina must be tall andthin just because she happens to be thin, you might reasonably objectby pointing out the possibility that Tina may not be tall. Model M1highlights this possibility.

DIRECT COMPOSITIONALITYSo far we have been paying little attention to sentence structure.However, as mentioned in the introduction, one of our main interestsis how meanings of natural language expressions are related to thesyntactic forms of these expressions. For instance, let us considerthe following two sentences:

(2.12) a. All pianists are composers, and Tina is a pianist.b. All composers are pianists, and Tina is a pianist.



A. B.

S

Tina is AP

tall and thin

IS(tina, AND(tall, thin))

tina IS AND(tall, thin)

tall AND thin

Figure 2.3 Syntactic structure and compositional derivation of denota-tions for Tina is tall and thin.

Sentences (2.12a) and (2.12b) contain the same words but in adifferent order. Consequently, the meanings of these two sentences arerather different. In particular, while (2.12a) entails the sentence Tina isa composer, sentence (2.12b) does not. The meaning of an expressionis not a soup made by simply putting together the meanings of words.Rather, the order of the words in a complex expression and thehierarchical structures that they form affect its meaning in systematicways. Since entailment relations between sentences reflect an aspectof their meanings, entailments are also sensitive to sentence structure.In the framework that we assume here, entailments are explainedby appealing to model-theoretic denotations. Therefore, the questionwe are facing is: how are syntactic structures used when definingdenotations of complex expressions? The general principle known ascompositionality provides an answer to this question. According tothis principle, the denotation of a complex expression is determined bythe denotations of its immediate parts and the ways they combine witheach other. For instance, in our analysis of the entailment (2.1)⇒(2.2),we treated the denotation of sentence (2.1) (Tina is tall and thin)as derived step by step from the denotations of its parts: the nameTina, the verb is, and the adjective phrase tall and thin. Figure 2.3summarizes our compositional analysis.Figure 2.3A shows the syntactic part-whole relations that we assume

for the sentence. In this structure we group together the string of wordstall and thin into one adjectival phrase, which we denote AP. Moregenerally, tree diagrams as in Figure 2.3A represent the sentence’sconstituents: the parts of the sentence that function as grammatical


MEANING AND FORM 29

units. In this case, besides the sentence itself and the words it contains,the only syntactic constituent assumed is the adjectival phrase talland thin. Figure 2.3B describes how denotations of constituents arederived from the denotations of their immediate parts. The denotationof the adjectival phrase tall and thin is determined by combining thedenotations of its immediate parts: tall, AND and thin. The denotationof the whole sentence is determined by the denotations of its parts:tina, IS, and AND(tall, thin). What we get as a result is the truth-valuedenotation in (2.11). The way in which this truth-value is derivedis sanctioned by the compositionality principle on the basis of thestructure in Figure 2.3A. Note that compositionality would not allowus to derive the truth-value in any other order. For instance, onthe basis of the structure in Figure 2.3A, we would not be able tocompositionally define the denotation of the whole sentence directlyon the basis of the denotations of the adjectives tall and thin. Thesewords are not among the sentence’s immediate parts. Therefore,according to the compositionality principle, they can only indirectlyaffect its denotation.Summarizing, we have adopted the following general principle, and

seen how we follow it in our analysis of the entailment (2.1)⇒(2.2).

Compositionality: The denotation of a complex expression is deter-mined by the denotations of its immediate parts and the ways theycombine with each other.

A word of clarification should be added here about the role ofsemantic formulas in our analysis. Consider for instance the formulaIS( tina, AND(tall,thin) ) that we derive in Figure 2.3B. This formula isnot a representation of some abstract meaning, independent of thesentence structure. To the contrary, this formula is almost com-pletely identical to the structure in Figure 2.3A, while adding onlythe necessary semantic details for describing how the denotation isderived for sentence (2.1). Most importantly, the formula specifies thefunction-argument relations between the denotations of the sentenceconstituents. Thus, the formula IS(tina, AND(tall, thin)) is simply thesyntactic bracketing [Tina is [tall and thin]] imposed by the tree inFigure 2.3A, with twomodifications: (i) symbols for words are replacedby symbols of their denotations; and (ii) symbols for denotations may



be shuffled around in order to follow the convention of puttingfunction symbols to the left of their arguments. This respects thehighly restrictive nature of compositional analysis: the process onlyrequires a syntactic structure, denotations of lexical items, and a wayto glue the denotations together semantically. The real “semanticaction” is within these three components of the theory, not withinthe semantic formulas we use. This version of compositionality issometimes referred to as direct compositionality . In this paradigm,denotations in themodel are directly derived from syntactic structures,with no intermediate level of semantic or logical representation.

STRUCTURAL AMBIGUITYDirect compositionality helps to clarify an important issue in linguistictheory: the phenomenon of structural ambiguity . Consider the follow-ing sentence:

(2.13) Tina is not tall and thin.

Let us consult our intuitions with respect to the following question:does (2.13) entail sentence (2.2) (=Tina is thin) or not? This ismuch harder to judge than in the case of the entailment (2.1)⇒(2.2).However, there is a common intuition that (2.13) entails (2.2), butonly under particular usages. A speaker who wishes to convey theentailment (2.13)⇒(2.2) can do so by stressing the prosodic boundaryafter the word tall:

(2.14) Tina is not tall, and thin.

Without such an intonational pattern, a speaker can also use (2.13)felicitously for describing a situation where Tina is not thin. Forinstance, if we tell Sue that Tina is tall and thin, she may use (2.13)for denying the assertion, by saying something like:

(2.15) Come on, that isn’t true! Tina is not tall and thin: although sheis indeed very tall, you couldn’t possibly think of her as thin!

In this reaction, the way in which Sue uses sentence (2.13) clearlyindicates that she does not consider it to entail sentence (2.2).


MEANING AND FORM 31

A. B.

S

Tina is AP

AP

not tall

and thin

S

Tina is AP

not AP

tall and thin

Figure 2.4 Structural ambiguity.

Because the two usages of sentence (2.13) show differences betweenits “entailment potential”, we say that it is ambiguous. The “commaintonation” in (2.14) disambiguates the sentence. Another way of dis-ambiguating sentence (2.13) is illustrated in (2.15), using the contextof our conversation with Sue. In the former disambiguation, sentence(2.13) is used for entailing (2.2); in the latter disambiguation, theentailment from (2.13) to (2.2) is blocked. We refer to the two possibleusages of sentence (2.13) as readings of the sentence. One readingentails (2.2), the other does not. Another way to describe the “tworeadings” intuition is to note that sentence (2.13) may be intuitivelyclassified as both true and false in the same situation. Consider asituation where Tina is absolutely not thin. Context (2.15) highlightsthat sentence (2.13) may be used as true in this situation. By contrast,(2.14) highlights that the sentence also has the potential of being falsein the same situation.The striking thing about the ambiguity of sentence (2.13) is the ease

with which it can be described when we assume the compositionalityprinciple. Virtually all syntactic theories analyze sentence (2.13) ashaving two different syntactic structures, as illustrated in Figure 2.4.A simple phrase structure grammar that generates the structuralambiguity in Figure 2.4 is given in (2.16) below:

(2.16) AP−→ tall, thin, ...AP−→ AP and APAP−→ not AP



In words: an adjective phrase (AP) can be a simple adjective, or aconjunction of two other APs, or a negation of another AP. These rulesderive both structures in Figure 2.4. When a grammar generates morethan one structure for an expression in this way, we say that it treats theexpression as structurally ambiguous. You may think that the syntacticambiguity in Figure 2.4 is by itself already an elegant account of thesemantic ambiguity in (2.13). However, there is a gap in this account:why does it follow that the structural ambiguity of sentence (2.13)also makes it semantically ambiguous? Compositionality provides themissing link. When the two structures in Figure 2.4 are composition-ally analyzed, we immediately see that the same model may assignthem two different truth-values. Concretely, let us assume that thedenotation of the negation word not in (2.13) is the complementfunction, i.e. the function NOT that maps any subset A of E to itscomplement set:

NOT(A)= A= E−A= the set of all themembers of E that are not in A

Figure 2.5 uses the denotation NOT for illustrating the compositionalanalysis of the two structures in Figure 2.4. As Figure 2.5 shows, thecompositional process works differently for each of the two structuralanalyses of sentence (2.13). For each of the denotations in Figures 2.5Aand 2.5B to be 1, different requirements have to be satisfied. This isspecified in (2.17a) and (2.17b) below:

(2.17) a. IS(tina, AND(NOT(tall), thin))= 1This holds if and only if (iff) tina ∈ tall∩ thin.

b. IS(tina,NOT(AND(tall, thin)))= 1This holds if and only if tina ∈ tall∩ thin.

For the denotation in Figure 2.5A to be 1, the requirement in (2.17a)must hold. When this is the case, the entity tina is in the set thin,hence the truth-value assigned to the sentence Tina is thin (=(2.2))is also 1. Thus, our compositional analysis of the structure in Figure2.4A captures the reading of sentence (2.13) that entails sentence (2.2).By contrast, the denotation (2.17b) that is derived in Figure 2.5B doesnot guarantee that the entity tina is in the set thin. This is because theentity tinamay be in the complement set tall∩ thin while being in theset thin, as long as it is not in the set tall. Specifically, consider a model


MEANING AND FORM 33

Figure 2.5 Compositionality and ambiguity.

M where the entities tina, mary and john are t, m and j, respectively.Suppose further that the model M assigns the following denotations tothe adjectives thin and tall:

thin= {t, j} tall= {m, j}The model M represents a situation where Tina is thin but not tall,Mary is tall but not thin, and John is both thin and tall. In this model,the denotation in Figure 2.5B is the truth-value 1, but sentence (2.2)denotes the truth-value 0. This means that our compositional analysisof the structure in Figure 2.4B captures the reading of sentence (2.13)that does not entail sentence (2.2).In compositional systems, the structure that we assume for a sen-

tence strongly affects the entailment relations that our theory expectsfor it. When a sentence is assumed to be structurally ambiguous, a



compositional theory may assign different truth-values to its differentstructures. As a result, the theory may expect different entailment rela-tions to hold for the different structures. Accordingly, when speakersare confronted with such a sentence, they are expected to experiencewhat we informally call “semantic ambiguity”, i.e. some systematichesitations regarding some of the sentence’s entailments. Structuralambiguity is used as the basis of our account of semantic ambiguity.Once we have acknowledged the possibility of ambiguity, we preferto talk about the entailments that sentence structures show, and ofthe truth-values that are assigned to these structures. However, forthe sake of convenience, we often say that sentences themselves haveentailments and truth-values. This convention is harmless when thesentences in question are unambiguous, or when it is clear that we aretalking about a particular reading.Semanticists often distinguish the syntactic-semantic ambiguity of

sentences like (2.13) from another type of under-specification, whichis called vagueness. For instance, as we noted above, the sentence Tinais tall says little about Tina and her exact height. In some contexts, e.g.if Tina is known to be a Western female fashion model, the sentencemay be used for indicating that Tina is above 1.70 meters. In othercontexts, e.g. if Tina is known to be member of some community ofrelatively short people, the sentence may indicate that Tina is above1.50 meters. However, we do not consider these two usages as evidencethat the sentence must be assigned different structures with potentiallydifferent denotations. Rather, we say that the sentence Tina is tall isvague with respect to Tina’s height. Further specification of relevantheights is dealt with by augmenting our semantic treatment with apragmatic theory . Pragmatic theories also consider the way in whichsentences are used, and the effects of context on their use. Pragmatictheories aim as well to account for the way speakers resolve (or partlyresolve) vagueness in their actual use of language. Classical versionsof formal semantics did not aim to resolve vagueness, but currentsemantic theories often interact with pragmatics and describe the waycontext helps in resolving vagueness in actual language use.Vagueness is very prominent in the way natural languages are used,

and most sentences may be vague in one way or another. For instance,the sentence Sue is talking tells us nothing about Sue’s voice (loud orquiet, high or low, etc.), what Sue is talking about, who the addresseeis, etc. However, upon hearing the sentence, we may often use the


MEANING AND FORM 35

context to infer such information. For instance, suppose that we areat a conference and know that Sue is one of the speakers. In such acontext, we may draw some additional conclusions about the subjectof Sue’s talk and the addressees. Hearers often use context in thisway to extract more information from linguistic expressions, andspeakers often rely on their hearers to do that. In distinction fromentailment, such inferential processes which are based on contextualknowledge are defeasible. For instance, even when the context specifiesa particular conference where Sue is a speaker, wemay use the sentenceSue is talking to indicate that Sue is talking to a friend over thephone. What we saw in sentence (2.13) is quite different from thedefeasible reasoning that helps speakers and hearers in their attemptsto resolve vagueness of language utterances. The comma intonationin (2.14) illustrated that one phonological expression of sentence(2.13) indefeasibly entails sentence (2.2). This convinced us that bothstructures that we assigned to sentence (2.13) are semantically useful.The theoretical consideration was the key for our treatment of thesentence as semantically ambiguous, more than any “pure” linguisticintuition. Most decisions between ambiguity and vagueness involvesimilar theoretical considerations, rather than the direct judgments ofa speaker’s linguistic intuitions.

FURTHER READINGIntroductory: For methodological aspects of logical semantics, in-cluding truth-values, entailment and compositionality, see Gamut(1982, vol. 1). For more examples and discussion of structuralambiguity, see Zimmermann and Sternefeld (2013, ch. 3), and,in relation to vagueness, Kennedy (2011). For further discussionof compositionality, see Partee (1984). On defeasible reasoning,see Koons (2014). Levinson (1983) is an introductory textbook onpragmatics. On lexical semantics, see Cruse (1986); Murphy (2010).Meaning relations between words and concepts they refer to areextensively studied in the literature on categorization. See Laurenceand Margolis (1999); Smith (1988); Taylor (1989) for introductionsof these topics.

Advanced: The idea that sentences denote truth-values, and moregenerally, propositions (cf. Chapter 6), was proposed as central for



communication (Austin 1962; Searle 1969). The centrality of entail-ment and the model-theoretic TCC was also highlighted in seman-tic theories of non-indicative sentences, especially interrogatives(Groenendijk and Stokhof 1984, 2011). An alternative to themodel-theoretic approach to entailment is proof-theoretic semantics(Schroeder-Heister, 2014). In its application to natural language,proof-theoretic approaches are sometimes referred to as naturallogic. Some examples of work in this area are McAllester andGivan (1992); Sánchez (1991); Moss (2010). Defeasible reasoning inlanguage is related to common sense reasoning in work in artificialintelligence (Brewka et al. 1997) and cognitive psychology (Stenningand van Lambalgen 2007; Adler and Rips 2008). For more onpragmatic theories, and specifically the notion of implicature , seeGrice (1975); Geurts (2010); Chierchia et al. (2012). Much of thiswork pays close attention to the meaning and use of the wordor (cf. the choice between ‘inclusive’ and ‘exclusive’ denotationsin Exercise 6). Direct compositionality in contemporary seman-tics of natural language was first illustrated in Montague (1970a).For further work on compositionality, see Montague (1970b);Janssen (1983); Janssen with Partee (2011); Barker and Jacobson(2007); Pagin and Westerståhl (2010); Werning et al. (2012).

EXERCISES (ADVANCED: 4, 5, 6, 7, 8, 9, 10)1. In the following pairs of sentences, make a judgment on whether

there is an entailment between them, and if so, in which of the twopossible directions. For directions in which there is no entailment,describe informally a situation that makes one sentence true andthe other sentence false. For example, in the pair of sentences (2.1)and (2.2), we gave the judgment (2.1)⇒(2.2), and supported thenon-entailment (2.2) �⇒(2.1) by describing a situation in whichTina is thin but not tall.(i) a. Tina got a B or a C. b. Tina got a B.(ii) a. Tina is neither tall nor thin. b. Tina is not thin.(iii) a. Mary arrived. b. Someone arrived.(iv) a. John saw fewer than four students. b. John saw no students.(v) a. The ball is in the room. b. The box is in the room and theball is in the box.(vi) a. Hillary is not a blond girl. b. Hillary is not a girl.


MEANING AND FORM 37

(vii) a.Hillary is a blond girl. b. Hillary is a girl.(viii) a. Tina is a Danish flutist and a physicist. b. Tina is a Danishphysicist and a flutist.(ix) a. Tina is not tall but taller than Mary. b. Mary is not tall.(x) a.Mary ran. b. Mary ran quickly.(xi) a. I saw fewer than five horses that ran. b. I saw fewer thanfive black horses that ran.(xii) a. I saw fewer than five horses that ran. b. I saw fewer thanfive animals that ran.(xiii) a. Exactly five pianists in this room are French composers. b.Exactly five composers in this room are French pianists.(xiv) a. No tall politician is multilingual. b. Every politician ismonolingual.(xv) a. No politician is absent. b. Every politician is present.(xvi) a. At most three pacifists are vegetarians. b. At most threevegetarians are pacifists.(xvii) a. All but at most three pacifists are vegetarians. b. At mostthree non-vegetarians are pacifists.

2. Each of the following sentences is standardly considered to bestructurally ambiguous. For each sentence suggest two structures,and show an entailment that one structure intuitively supports andthe other structure does not:(i) I read that Dan published an article in the newspaper.(ii) Sue is blond or tall and thin.(iii) The policeman saw the man with the telescope.(iv) Rich Americans and Russians like to spend money.(v) Sue told some man that Dan liked the story.(vi) Dan ate the lettuce wet.(vii) Sue didn’t see a spot on the floor.

3. Table 2.2 shows different denotations for the expressions in sen-tences (2.1) and (2.2) in different models. We used these modelsand the truth-values they assign to the sentences to support ourclaim that the TCC explains the entailment (2.1)⇒(2.2), and thenon-entailment (2.2) �⇒(2.1).The table on the right gives the expressions for the two analyses inFigure 2.5 of the sentence Tina is not tall and thin.

a. Add to this table the missing denotations of these expressionswithin the three models M1, M2 and M3.



b. Verify that the truth-values that you assigned to the two analy-ses in Figure 2.5 support the intuition that the analysis inFigure 2.4A entails sentence (2.2), whereas the analysis inFigure 2.4B does not entail (2.2).

Expression Denotations in example models withE = {a, b, c, d}M1 M2 M3

Tina a b btall {b, c} {b, d} {a, b, d}thin {a, b, c} {b, c} {a, c, d}not tall[not tall] and thinTina is [[not tall] and thin]tall and thinnot [tall and thin]Tina is [not [tall and thin]]Tina is thin

4.a. Mark the pairs of sentences in Exercise 1 that you consideredequivalent.

b. Give three more examples for pairs of sentences that you con-sider intuitively equivalent.

c. State the formal condition that a semantic theory that satisfiesthe TCC has to satisfy with respect to equivalent sentences.

5. Consider the ungrammaticality of the following strings of words.(i) *Tina is both tall *Tina is both not tall *Tina is both tall orthinTo account for this ungrammaticality, let us assume that the wordboth only appears in adjective phrases of the structure both AP1and AP2. Thus, a constituent both X is only grammatical when Xis an and-conjunction of two adjectives or adjective phrases; hencethe grammaticality of the string both tall and thin as opposed to theungrammaticality of the strings in (i), where X is tall, not tall andtall or thin, respectively. We assume further that the denotationof a constituent both AP1 and AP2 is the same as the denota-tion of the parallel constituent AP1 and AP2 as analyzed in thischapter.


MEANING AND FORM 39

a. With these syntactic and semantic assumptions, write down thedenotations assigned to the following sentences in terms of thedenotations tina, tall, thin, IS and AND.(ii) Tina is both not tall and thin. (iii) Tina is not both tall andthin.

b. Explain why the denotations you suggested for (ii) and (iii)account for the (non-)entailments (ii)⇒(2.2) and (iii) �⇒(2.2).

c. Consider the equivalence between the following sentences:(iv) Tina is both not tall and not thin. (v) Tina is neither tallnor thin.Suggest a proper denotation for the constituent neither tall northin in (v) in terms of the denotations tall and thin (standingfor sets of entities). Explain how the denotation you suggest,together with our assumptions in items 5a and 5b above, explainthe equivalence (iv)⇔(v).

6. Consider the following sentence:(i) Tina is [tall or thin].The inclusive or exclusive analyses for the coordinator or in-volve denotations that employ the union and symmetric differencefunctions, respectively – the functions defined as follows for allA, B ⊆ E :

ORin(A, B)= A∪ B= the set of members of E that are in A, in B

or in both A and B

ORex(A, B) = (A− B)∪ (B − A)= (A∪ B)− (A∩ B)= the set of members of E that are in A or B , but not

both A and B

Consider the following sentential structures:(ii) Tina is not [tall and thin].(iii) Tina is not [tall or thin].(iv) Tina is [not tall] and [not thin].(v) Tina is [not tall] or [not thin].

a. Assuming that the word or denotes the function ORin, write downall the entailments that the TCC expects in (ii)–(v). Answer thesame question, but now assuming that or denotes ORex.



b. Which of the two entailment patterns in 6a better captures yourlinguistic intuitions about (ii)–(v)?

c. Under one of the two analyses of or, one of the structures (ii)–(v) is expected to be equivalent to (i). Which structure is it, andunder which analysis? Support your answer by a set-theoreticalequation.

7. A pair of sentences (or readings/structures) is said to be treated ascontradictory if, whenever one of the sentences is taken to denote 1,the other denotes 0. For instance, under the analysis in this chapter,the sentencesMary is tall andMary is not tall are contradictory.

a. Give more examples for contradictory pairs of sentences/structures under the assumptions of this chapter.

b. Consider the sentences The bottle is empty and The bottle isfull. Suggest a theoretical assumption that would render thesesentences contradictory.

c. Give an entailment that is accounted for by the same assump-tion.

d. Show that according to our account, the denotation of thesentence Tina is tall and not tall is 0 in any model. Such asentence is classified as a contradiction. Show more examples forsentences that our account treats as contradictions.

e. Show that according to both our treatments of or in Exercise 6,the denotation of the sentence Tina is tall or not tall is 1 in anymodel. Such sentences are classified as tautological . Show moreexamples for sentences that our account treats as tautological.

f. Show that the TCC expects that any contradictory sentenceentails any sentence in natural language, and that any tautologyis entailed by any sentence in natural language. Does this expec-tation agree with your linguistic intuitions? If it does not, do youhave ideas about how the problem can be solved?

8. We assume that entailments between sentences (or structures)have the following properties.Reflexivity: Every sentence S entails itself.Transitivity: For all sentences S1, S2, S3: if S1 entails S2 and S2

entails S3, then S1 entails S3.Reflexivity and transitivity characterize entailments as a preorderrelation on sentences/structures.


MEANING AND FORM 41

Consider the following entailments:(i) Tina is tall, and Ms. Turner is neither tall nor thin⇒ Tina istall, and Ms. Turner is not tall.(ii) Tina is tall, and Ms. Turner is neither tall nor thin⇒ Tina isnot Ms. Turner.Show an entailment that illustrates transitivity together with en-tailments (i) and (ii).

9. Consider the following structurally ambiguous sentence (=(ii)from Exercise 2).(i) Tina is blond or tall and thin.a. For sentence (i), write down the denotations derived for the two

structures using the inclusive denotation of or from Exercise 6,and the denotations tina, blond, tall and thin.

b. Give specific set denotations for the words blond, tall and thinthat make one of these denotations true (1), while making theother denotation false (0).

c. Using the both . . . and construction from Exercise 4, find twounambiguous sentences, each of which is equivalent to one ofthe structural analyses you have given for sentence (i).

d. Under an inclusive interpretation of or, which of the two sen-tences you found in 9c is expected to be equivalent to thefollowing sentence?(ii) Tina is both blond and thin or both tall and thin.

e. Write down the set-theoretical equation that supports this equiv-alence.

f. Using our assumptions in this chapter, find a structurally am-biguous sentence whose two readings are analyzed as equivalent.

10. Consider the following entailment:(i) Tina has much money in her bank account, and Bill has onecent less than Tina in his bank account⇒ Bill has much money inhis bank account.a. We adopt the following assumption: Tina has m cents in her

bank account, wherem is some positive natural number. Further,we assume that entailment is transitive. Show that with theseassumptions, you can use the entailment pattern in (i) to supportan entailment with the following contradictory conclusion: Ms.X has much money in her bank account, and Ms. X has no moneyin her bank account.



b. The ability to rely on transitivity of entailments to support suchabsurd conclusions is known as the Sorites Paradox. Suggest apossible resolution of this paradox by modifying our assump-tions in 10a and/or our assumption that entailment relations aretransitive.

SOLUTIONS TO SELECTED EXERCISES3.

ExpressionDenotations in examplemodels withE = {a, b, c, d}M1 M2 M3

Tina a b btall {b, c} {b, d} {a, b, d}thin {a, b, c} {b, c} {a, c, d}not tall {a, d} {a, c} {c}[not tall] and thin {a} {c} {c}Tina is [[not tall] and thin] 1 0 0tall and thin {b, c} {b} {a, d}not [tall and thin] {a, d} {a, c, d} {b, c}Tina is [not [tall and thin]] 1 0 1Tina is thin 1 1 0

4.c. In any theory T that satisfies the TCC, sentences S1 and S2 areequivalent if and only if for all models M in T , [[S1]]M=[[S2]]M .

5. a–b. The truth-values and the accounts of the (non-)entail-ments are identical to the truth-values for the ambiguoussentence (2.13) and the corresponding (non-)entailment from(2.13) to (2.2).

c. [[neither tall nor thin]] = tall∩ thin=AND(NOT(tall),NOT(thin))= [[both not tall and not thin]]

6.a. ORin: (iii)⇒(ii); (iv)⇒(ii); (ii)⇔(v); (iii)⇔(iv); (iii)⇒(v);(iv)⇒(v).

ORex: (iv)⇒(ii); (v)⇒(ii); (iv)⇒(iii).c. (v); the ORex analysis; (A− B)∪ (B − A)= (B − A)∪ (A−

B)= (A− B)∪ (B − A).


MEANING AND FORM 43

7.a. Mary is neither tall nor thin, Mary is tall or thin; Mary is tall andnot thin, Mary is thin; Mary is [not tall] or [not thin], Mary istall and thin.

b. The adjectives empty and full denote disjoint sets: empty∩full= ∅.

c. The bottle is empty⇒ The bottle is not full; The bottle is full⇒The bottle is not empty.

8. Tina is tall, and Ms. Turner is not tall⇒ Tina is not Ms. Turner(=(2.3a)).

9.a. IS(tina,AND(ORin(blond,tall),thin))IS(tina,ORin(blond,AND(tall,thin)))

b. blond= {tina}; tall= thin= ∅c. Tina is both blond or tall and thin.

Tina is blond or both tall and thin.d. Tina is both blond or tall and thin.e. (A∪ B)∩C = (A∩C )∪ (B ∩C )f. Tina is blond and tall and thin

10.a. Consider the following general entailment scheme, based on (i):(i ′)Ms. n hasmuchmoney inMs. n’s bank account andMs. n+ 1has one cent less thanMs. n inMs. n+ 1’s bank account ⇒ Ms.n+ 1 has much money inMs. n+ 1’s bank account.We can deduce from (i ′), by induction on the transitivity ofentailment, that the following (unacceptable) entailment is in-tuitively valid:Ms. 1 has much money in her bank account, and Ms. 1 has mcents in her bank account ⇒ Ms. m+ 1 has much money in herbank account, andMs. m+ 1 has no cents in her bank account.

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Elements of Formal Semantics - Universiteit Utrechtyoad/efs/EFS-ch2-online.pdf · Elements of Formal Semantics introduces some of the foundational concepts, principles and techniques